On obtaining higher order convergence for smooth periodic functions
type: Item
instance of: scholarly article
title: On obtaining higher order convergence for smooth periodic functions
zbMATH Open document ID: 1151.65019
author: Bart Vandewoestyne
author: Ronald Cools
published in: Journal of Complexity
publication date: 2008-07-21T00:00:00Z
review text: Consider the problem of approximating integrals of the form \[ I=\int_{[0,1)^{s}}f(x)\,dx, \] where \(f\) is a function on the \(s\)-dimensional unit cube \([0,1)^{s}\) with an absolutely convergent Fourier series. Of particular interest are cubature rules of the form \[ Q_{N}=\sum_{k=1}^{N}w_{k}f(x_{k}), \] with sample points of an infinite sequence \(x_{1}, x_{2}, \dots\) and weights \(w_{k}\). The authors show how to construct a general and open algorithm for \textit{H. Niederreiter}'s method [Aequationes Math. 8 (1972), 304--311 (1971; Zbl 0252.65023); Diophantine Approx. Appl., Proc. Conf. Washington 1972; 129--199 (1973; Zbl 0268.65014)] and for \textit{M. Sugihara} and \textit{K. Murota}'s [Math. Comput. 39, 549--554 (1982; Zbl 0502.65009)] method, so that one can use them to obtain general higher order error convergence for smooth periodic functions. The memory cost of these algorithms is low and they are linear in the number of steps. The absolute errors as a function of the number of sample points for both methods are comparable. Recall that Niederreiter considers the following algorithm \[ Q_{N}^{(q)}=N^{-q}\sum_{n=0}^{q(N-1)}a_{N,n}^{(q)}f(\{n\alpha\}), \] where \(\alpha\) is an \(s\)-dimensional vector of irrational components, \(\{\cdot\}\) is the fractional parts and the weights \(a_{N,n}^{(q)}\) are uniquely determined from some polynomial identity. Sugihara and Murota consider the algorithm \[ \widetilde{Q}_{\widetilde{N}}^{(q)}=\widetilde{N}^{-1}\sum_{k=1}^{\widetilde{N}}w_{q} \left(\frac{k}{\widetilde{N}}\right)f(\{k\alpha\}), \] where \(w_{q}(x)=A_{q}x^{q}(1-x)^{q}\) with \(A_{q}=\frac{(2q+1)!}{(q!)^{2}}\).
reviewed by: Ana Maria Acu
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 65C05
zbMATH DE Number: 5303160
zbMATH Keywords: multivariate numerical integration
zbMATH Keywords: cubature
zbMATH Keywords: periodic functions
zbMATH Keywords: quasi-Monte Carlo method
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
Wikidata QID: Q57778870
full work available at URL: j.jco.2007.10.002
OpenAlex ID: W2061874121
cites work: Explicit Constructions of Quasi-Monte Carlo Rules for the Numerical Integration of High-Dimensional Periodic Functions
cites work: On a number-theoretical integration method
cites work: Q3214067
cites work: A Note on Haselgrove's Method for Numerical Integration
cites work: On obtaining quadratic and cubic error convergence using weighted Kronecker-sequences
DOI: 10.1016/J.JCO.2007.10.002
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description: scientific article; zbMATH DE number 5303160